This review discusses the essential nature of edge-flames, how they are defined, the nature of their structure, how it is that both positive and negative edge-speeds can arise. Their roots in mixing-layer flames are extensively described, but premixed edge-flames are also discussed, rooted in the multivalued ID solutions that can be exhibited by certain kinds of deflagrations. Cellular instabilities of both non-premixed and premixed edge-flames located in counterflows are examined, and links with flame-strings and flame-balls are discussed when one or more Lewis numbers are small. And the pulsating instability commonly associated with large Lewis numbers is examined in a simple model context that has some of the characteristics of experimentally realizable flames, such as the microgravity candle flame. Lifted edge-flames, both laminar and turbulent are described, as are some aspects of non-premixed edge-flame holding. (C) 2002 Elsevier Science Ltd. All rights reserved.